Structural and Multidisciplinary Optimization

, Volume 58, Issue 6, pp 2533–2544 | Cite as

Robust topology optimization of skeletal structures with imperfect structural members

  • Babak Ahmadi
  • Mehdi Jalalpour
  • Alireza Asadpoure
  • Mazdak TootkaboniEmail author


A topology optimization framework is proposed for robust design of skeletal structures with stochastically imperfect structural members. Imperfections are modeled as uncertain members’ out-of-straightness using curved frame elements in the form of predefined functions with random magnitudes throughout the structure. The stochastic perturbation method is used for propagating the imperfection uncertainty up to the structural response level, and the expected value of performance measure or constraint is used to form the stochastic topology optimization problem. Sensitivities are derived explicitly using the adjoint method and are used in conjunction with an efficient gradient-based optimizer in search for robust optimal topologies. Topological designs for three representative examples are investigated with the proposed algorithm and the resulting topologies are compared with the deterministic designs. It is observed that the new designs primarily feature load path diversification, which is pronounced with increasing level of uncertainty, and occasionally member thickening to mitigate the impact of the uncertainty in members’ out-of-straightness on structural performance.


Topology optimization Gradient-based optimizer Adjoint method Structural imperfections Stochastic perturbation Curved beam element 



This work was supported by the National Science Foundation under Grant No. CMMI-1401575. This support is gratefully acknowledged. Asadpoure also acknowledges support from University of Massachusetts College of Engineering.


  1. Allaire G, Jouve F, Toader AM (2004) Structural optimization using sensitivity analysis and a level-set method. J Comput Phys 194(1):363–393MathSciNetCrossRefGoogle Scholar
  2. Allaire G, Dapogny C (2014) A linearized approach to worst-case design in parametric and geometric shape optimization. Math Models Methods Appl Sci 24(11):2199–2257MathSciNetCrossRefGoogle Scholar
  3. Amir O, Sigmund O, Lazarov BS, Schevenels M (2012) Efficient reanalysis techniques for robust topology optimization. Comput Methods Appl Mech Eng 245–246:217–231MathSciNetCrossRefGoogle Scholar
  4. Asadpoure A, Tootkaboni M, Guest JK (2011) Robust topology optimization of structures with uncertainties in stiffness–application to truss structures. Comput Struct 89(11):1131–1141CrossRefGoogle Scholar
  5. Ben-Tal A, Nemirovski A (1997) Robust truss topology design via semidefinite programming. SIAM J Optim 7(4):991–1016MathSciNetCrossRefGoogle Scholar
  6. Bendsoe MP, Sigmund O (2004) Topology optimization: theory, methods and applications. Springer, BerlinCrossRefGoogle Scholar
  7. Cambou B (1975) Application of first-order uncertainty analysis in the finite element method in linear elasticity. In: Proceedings of 2nd international conference on applications of statistics and probability in soil and structural engineering, pp 67–87Google Scholar
  8. Changizi N, Jalalpour M (2017a) Robust topology optimization of frame structures under geometric or material properties uncertainties. Struct Multidiscip Optim 51(4):1–17MathSciNetGoogle Scholar
  9. Changizi N, Kaboodanian H, Jalalpour M (2017b) Stress-based topology optimization of frame structures under geometric uncertainty. Comput Methods Appl Mech Eng 315(2):121–140. MathSciNetCrossRefGoogle Scholar
  10. Chen S, Chen W, Lee S (2010) Level set based robust shape and topology optimization under random field uncertainties. Struct Multidiscip Optim 41(4):507–524MathSciNetCrossRefGoogle Scholar
  11. Collins JD, Thomson WT (1969) The eigenvalue problem for structural systems with statistical properties. AIAA J 7(4):642–648CrossRefGoogle Scholar
  12. Csébfalvi A (2014) A new theoretical approach for robust truss optimization with uncertain load directions. Mech Based Des Struct Mach 42(4):442–453CrossRefGoogle Scholar
  13. Deaton JD, Grandhi RV (2014) A survey of structural and multidisciplinary continuum topology optimization: post 2000. Struct Multidiscip Optim 49(1):1–38MathSciNetCrossRefGoogle Scholar
  14. Dunning PD, Kim HA (2013) Robust topology optimization: minimization of expected and variance of compliance. AIAA J 51(11):2656–2663CrossRefGoogle Scholar
  15. Gu X, Sun G, Li G, Mao L, Li Q (2013) A comparative study on multiobjective reliable and robust optimization for crashworthiness design of vehicle structure. Struct Multidiscip Optim 48(3):669–684CrossRefGoogle Scholar
  16. Guest J, Igusa T (2008) Structural optimization under uncertain loads and nodal locations. Comput Meth Appl Mech Eng 198(1):116–124MathSciNetCrossRefGoogle Scholar
  17. Guo X, Bai W, Zhang W, Gao X (2009) Confidence structural robust design and optimization under stiffness and load uncertainties. Comput Methods Appl Mech Eng 198(41-44):3378–3399MathSciNetCrossRefGoogle Scholar
  18. Guo X, Du J, Gao X (2011) Confidence structural robust optimization by non-linear semidefinite programming-based single-level formulation. Int J Numer Methods Eng 86(8):953–974MathSciNetCrossRefGoogle Scholar
  19. Guo X, Zhang W, Zhong W (2014) Doing topology optimization explicitly and geometrically a new moving morphable components based framework. J Appl Mech 81(8):081,009CrossRefGoogle Scholar
  20. Hisada T, Nakagiri S (1981) Stochastic finite element method developed for structural safety and reliability. In: Proceedings of the 3rd international conference on structural safety and reliability, pp 395–408Google Scholar
  21. Jalalpour M, Igusa T, Guest JK (2011) Optimal design of trusses with geometric imperfections: accounting for global instability. Int J Solids Struct 48(21):3011–3019CrossRefGoogle Scholar
  22. Jalalpour M, Guest JK, Igusa T (2013) Reliability-based topology optimization of trusses with stochastic stiffness. Struct Saf 43 :41–49CrossRefGoogle Scholar
  23. Jalalpour M, Tootkaboni M (2016) An efficient approach to reliability-based topology optimization for continua under material uncertainty. Struct Multidiscip Optim 53(4):759–772MathSciNetCrossRefGoogle Scholar
  24. Jang GW, Dijk NP, Keulen F (2012) Topology optimization of mems considering etching uncertainties using the level-set method. Int J Numer Methods Eng 92(6):571–588MathSciNetCrossRefGoogle Scholar
  25. Jansen M, Lombaert G, Diehl M, Lazarov BS, Sigmund O, Schevenels M (2013) Robust topology optimization accounting for misplacement of material. Struct Multidiscip Optim 47(3):317–333MathSciNetCrossRefGoogle Scholar
  26. Jansen M, Lombaert G, Schevenels M (2015) Robust topology optimization of structures with imperfect geometry based on geometric nonlinear analysis. Comput Methods Appl Mech Eng 285:452–467., MathSciNetCrossRefGoogle Scholar
  27. Keshavarzzadeh V, Fernandez F, Tortorelli DA (2017) Topology optimization under uncertainty via non-intrusive polynomial chaos expansion. Comput Methods Appl Mech Eng 318:120–147., MathSciNetCrossRefGoogle Scholar
  28. Kleiber M, Hien TD (1992) The stochastic finite element method: basic perturbation technique and computer implementation. Wiley, New YorkzbMATHGoogle Scholar
  29. Lazarov BS, Schevenels M, Sigmund O (2012) Topology optimization with geometric uncertainties by perturbation techniques. Int J Numer Methods Eng 90(11):1321–1336CrossRefGoogle Scholar
  30. Liu WK, Belytschko T, Mani A (1986) Probabilistic finite elements for nonlinear structural dynamics. Comput Methods Appl Mech Eng 56(1):61–81CrossRefGoogle Scholar
  31. Lógó J (2007) New type of optimality criteria method in case of probabilistic loading conditions. Mech Based Des Struct Mach 35(2):147–162CrossRefGoogle Scholar
  32. Lógó J, Ghaemi M, Rad MM (2009) Optimal topologies in case of probabilistic loading: the influence of load correlation. Mech Based Des Struct Mach 37(3):327–348CrossRefGoogle Scholar
  33. Lombardi M, Haftka RT (1998) Anti-optimization technique for structural design under load uncertainties. Comput Methods Appl Mech Eng 157(1-2):19–31CrossRefGoogle Scholar
  34. Luo Z, Chen LP, Yang J, Zhang YQ, Abdel-Malek K (2006) Fuzzy tolerance multilevel approach for structural topology optimization. Comput Struct 84(3):127–140MathSciNetCrossRefGoogle Scholar
  35. Martínez-Frutos J, Herrero-Pérez D, Kessler M, Periago F (2016) Robust shape optimization of continuous structures via the level set method. Comput Methods Appl Mech Eng 305:271–291MathSciNetCrossRefGoogle Scholar
  36. Medina JC, Taflanidis A (2015) Probabilistic measures for assessing appropriateness of robust design optimization solutions. Struct Multidiscip Optim 51(4):813–834MathSciNetCrossRefGoogle Scholar
  37. Moens D, Vandepitte D (2005) A survey of non-probabilistic uncertainty treatment in finite element analysis. Comput Methods Appl Mech Eng 194(1216):1527–1555, special Issue on Computational Methods in Stochastic Mechanics and Reliability Analysis. CrossRefGoogle Scholar
  38. Richardson JN, Coelho RF, Adriaenssens S (2015) Robust topology optimization of truss structures with random loading and material properties: a multiobjective perspective. Comput Struct 154:41–47CrossRefGoogle Scholar
  39. Sandgren E, Cameron T (2002) Robust design optimization of structures through consideration of variation. Comput Struct 80(20):1605–1613CrossRefGoogle Scholar
  40. Schevenels M, Lazarov BS, Sigmund O (2011) Robust topology optimization accounting for spatially varying manufacturing errors. Comput Methods Appl Mech Eng 200(49):3613–3627CrossRefGoogle Scholar
  41. Shinozuka M, Astill CJ (1972) Random eigenvalue problems in structural analysis. AIAA J 10(4):456–462CrossRefGoogle Scholar
  42. Sigmund O, Jensen JS (2003) Systematic design of phononic band–gap materials and structures by topology optimization. Philos Trans Royal Soc Lond A: Math Phys Eng Sci 361(1806):1001–1019MathSciNetCrossRefGoogle Scholar
  43. Sigmund O (2009) Manufacturing tolerant topology optimization. Acta Mech Sinica 25(2):227–239CrossRefGoogle Scholar
  44. Sigmund O (2011) On the usefulness of non-gradient approaches in topology optimization. Struct Multidiscip Optim 43(5):589–596MathSciNetCrossRefGoogle Scholar
  45. Suzuki K, Kikuchi N (1991) A homogenization method for shape and topology optimization. Comput Methods Appl Mech Eng 93(3):291–318CrossRefGoogle Scholar
  46. The MathWorks Inc (2017) MATLAB - Optimization Toolbox, Version 7.6. The MathWorks Inc., Natick, Massachusetts,
  47. Tootkaboni M, Asadpoure A, Guest JK (2012) Topology optimization of continuum structures under uncertainty–a polynomial chaos approach. Comput Methods Appl Mech Eng 201:263–275MathSciNetCrossRefGoogle Scholar
  48. Venini P, Pingaro M (2017) An innovative h-norm based worst case scenario approach for dynamic compliance optimization with applications to viscoelastic beams. Struct Multidiscip Optim 55(5):1685–1710MathSciNetCrossRefGoogle Scholar
  49. Wu J, Aage N, Westermann R, Sigmund O (2018) Infill optimization for additive manufacturingapproaching bone-like porous structures. IEEE Trans Vis Comput Graph 24(2):1127–1140CrossRefGoogle Scholar
  50. Zhang W, Yuan J, Zhang J, Guo X (2016) A new topology optimization approach based on moving morphable components (mmc) and the ersatz material model. Struct Multidiscip Optim 53(6):1243–1260MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Babak Ahmadi
    • 1
  • Mehdi Jalalpour
    • 2
  • Alireza Asadpoure
    • 1
  • Mazdak Tootkaboni
    • 1
    Email author
  1. 1.Department of Civil and Environmental EngineeringUniversity of MassachusettsDartmouthUSA
  2. 2.Department of Civil and Environmental EngineeringCleveland State UniversityClevelandUSA

Personalised recommendations